1
00:00:00,260 --> 00:00:01,778
Centered around A.

2
00:00:01,778 --> 00:00:02,433
[SOUND]

3
00:00:02,433 --> 00:00:08,197
Well thus far, we've been talking about
Maclaurin

4
00:00:08,197 --> 00:00:14,880
series or about Taylor series centered
around zero.

5
00:00:14,880 --> 00:00:18,200
But, I can write down a power series for a
function,

6
00:00:18,200 --> 00:00:22,080
not centered around zero, but centered
around some other point A.

7
00:00:22,080 --> 00:00:28,540
Well what I mean is I could try to write
the function f as a power series.

8
00:00:28,540 --> 00:00:33,100
N goes from zero to infinity, some
coefficient C sub n, not just

9
00:00:33,100 --> 00:00:38,390
times x to the n, but times x minus a to
the nth power.

10
00:00:38,390 --> 00:00:42,490
Well, just like before, let's assume that
f has a power series

11
00:00:42,490 --> 00:00:48,110
representation, and then try to figure out
what those coefficients must be.

12
00:00:48,110 --> 00:00:49,800
I want to be a little bit more precise,
right.

13
00:00:49,800 --> 00:00:53,950
What I'm saying here is I'm going to
assume this holds whenever x

14
00:00:53,950 --> 00:00:56,510
is, say, within big R of a.

15
00:00:57,520 --> 00:01:00,080
And assuming this is true, I want to
figure out what

16
00:01:00,080 --> 00:01:04,020
these coefficients have to be in terms of
the function, f.

17
00:01:04,020 --> 00:01:05,890
So I'm just going to write down the first
few terms, right.

18
00:01:05,890 --> 00:01:11,160
When I plug in n equals 0, I just get C
sub 0 times 1, because it's something to

19
00:01:11,160 --> 00:01:18,960
the 0th power, plus when I plug in n
equals 1, I get C sub 1 times x minus a to

20
00:01:18,960 --> 00:01:20,530
the 1st power.

21
00:01:20,530 --> 00:01:26,710
Plus, when I plug in n equals 2, it's C
sub 2 times x minus a squared.

22
00:01:26,710 --> 00:01:32,820
And then it keeps on going.
Now, what happens when I plug in a for x?

23
00:01:32,820 --> 00:01:36,910
So that's asking, what is f of a?

24
00:01:36,910 --> 00:01:40,480
Well in that case it's C sub zero plus
what's this term?

25
00:01:40,480 --> 00:01:43,840
It's just zero plus what's this term?
Well it's just zero plus,

26
00:01:43,840 --> 00:01:45,770
all the other terms are zero.

27
00:01:45,770 --> 00:01:50,180
So this is telling me that C sub zero is
equal to f of a.

28
00:01:50,180 --> 00:01:54,800
Or another way to say that, is if I knew
that this was true for values

29
00:01:54,800 --> 00:01:59,690
of x within big R of a, then I know what C
sub zero has to be.

30
00:01:59,690 --> 00:02:03,930
C sub zero has to be the value of f at the
point a.

31
00:02:03,930 --> 00:02:06,430
The next coefficient can be computed in
terms

32
00:02:06,430 --> 00:02:09,600
of the derivative, but not the derivative
at zero,

33
00:02:09,600 --> 00:02:12,170
but the derivative at the point a.

34
00:02:12,170 --> 00:02:14,100
Well let's differentiate this, see what we
get.

35
00:02:14,100 --> 00:02:18,840
So I'm assuming that this is true whenever
x is within big R of a.

36
00:02:18,840 --> 00:02:22,450
So the same will be true for the
derivative, at least, so the derivative of

37
00:02:22,450 --> 00:02:28,080
f of x will be the sum, n goes not from 0
but from 1 to infinity of the derivative

38
00:02:28,080 --> 00:02:35,320
of this, which is C sub n times n times x
minus a to the n minus 1, and this

39
00:02:35,320 --> 00:02:39,620
is true at least when x is within that
same big R of a.

40
00:02:39,620 --> 00:02:41,540
Okay, let me write down the first few
terms here.

41
00:02:41,540 --> 00:02:43,130
Let's plug in n equals 1.

42
00:02:43,130 --> 00:02:47,580
I get just C sub 1 times 1 times x minus a
to the 0th power.

43
00:02:47,580 --> 00:02:51,500
So that's just C sub 1, plus when I plug
in n equals 2.

44
00:02:51,500 --> 00:02:56,720
I get C sub 2 times 2 times x minus a to
the 1st power.

45
00:02:57,730 --> 00:03:00,612
When I plug in n equals 3, I get C sub 3

46
00:03:00,612 --> 00:03:06,440
times 3 times x minus a squared, and then
it keeps on going.

47
00:03:06,440 --> 00:03:10,540
And now note what happens when I plug in a
for x.

48
00:03:10,540 --> 00:03:13,590
Alright?
What's f prime of a?

49
00:03:13,590 --> 00:03:15,000
Well it's just like what happened before!

50
00:03:15,000 --> 00:03:17,868
I've got C sub 1, but then this C sub

51
00:03:17,868 --> 00:03:20,920
2 times 2 times x minus a, that term
vanishes.

52
00:03:20,920 --> 00:03:23,250
This term here also has an x minus a, so
when

53
00:03:23,250 --> 00:03:25,840
x is equal to a, this term vanishes, all
the later

54
00:03:25,840 --> 00:03:27,320
terms vanish.

55
00:03:27,320 --> 00:03:29,920
This means that I can recover the value of
the C

56
00:03:29,920 --> 00:03:34,960
sub 1 coefficient, just by differentiating
the function at the point a.

57
00:03:34,960 --> 00:03:36,310
And so on, right.

58
00:03:36,310 --> 00:03:41,680
Just like before, I can now write down a
formula for the nth coefficient.

59
00:03:41,680 --> 00:03:46,130
So in general, what I'll find is that C
sub n can be

60
00:03:46,130 --> 00:03:51,260
computed by taking the nth derivative of f
at the point a and dividing

61
00:03:51,260 --> 00:03:55,170
that by n factorial.
So let me summarize all of this.

62
00:03:55,170 --> 00:03:59,820
The Taylor series for function f centered
at a, or sometimes you'll see

63
00:03:59,820 --> 00:04:04,660
just the Taylor series around a, for the
function f, it's given by this.

64
00:04:04,660 --> 00:04:07,950
It's the sum n goes from zero to infinity,
of the nth

65
00:04:07,950 --> 00:04:12,250
derivative of f at a, that's the big
difference, divided by n factorial.

66
00:04:12,250 --> 00:04:15,540
Here's another difference.
x minus a to the nth power.

67
00:04:15,540 --> 00:04:16,300
I mean it

68
00:04:16,300 --> 00:04:16,910
sort of makes sense.
Right.

69
00:04:16,910 --> 00:04:20,670
The Taylor series around a, is a power
series around a.

70
00:04:20,670 --> 00:04:23,480
And so it's got this x minus a to the nth
term.

71
00:04:23,480 --> 00:04:25,870
But instead of calculating the derivative
at zero

72
00:04:25,870 --> 00:04:27,945
we're calculating the derivative at the
point a.

73
00:04:27,945 --> 00:04:37,945
[NOISE]